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1
Content available Analysis of the pointwise completeness and the pointwise degeneracy of the standard and fractional descriptor linear systems and electrical circuits
Kaczorek Tadeusz
Journal of Automation, Electronics and Electrical Engineering
|
2023
|
Vol. 5, No. 1
8--19
EN
The Drazin inverse of matrices is applied to analysis of the pointwise completeness and the pointwise degeneracy of the descriptor standard and fractional linear continuous-time and discrete-time systems. It is shown that: 1) The descriptor linear continuous-time system is pointwise complete if and only if the initial and final states belong to the same subspace. 2) The descriptor linear discrete-time system is not pointwise complete if its system matrix is singular. 3) System obtained by discretization of continuous-time system is always not pointwise complete. 4) The descriptor linear continuous-time system is not pointwise degenerated in any nonzero direction for all nonzero initial conditions. 5) The descriptor fractional system is pointwise complete if the matrix defined by (36) is invertible. 6) The descriptor fractional system is pointwise degenerated if and only if the condition (41) is satisfied. Considerations are illustrated by examples of descriptor linear electrical circuits.
2
Content available Positivity and cyclicity of descriptor electrical circuits with chain structure
Kaczorek Tadeusz, Borawski Kamil
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2022
|
Vol. 70, nr 1
art. no. e139955
EN
The positivity and cyclicity of descriptor linear electrical circuits with chain structure is considered. Two classes of descriptor linear electrical circuits are analyzed. Some new properties of these classes of electrical circuits are established. The results are extended to fractional descriptor linear electrical circuits.
3
Content available A new extension of the Cayley-Hamilton theorem to fractional different orders linear systems
Kaczorek Tadeusz
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2022
|
Vol. 70, nr 2
art. no. e139960
EN
The classical Cayley–Hamilton theorem is extended to fractional different order linear systems. The new theorems are applied to different orders fractional linear electrical circuits. The applications of new theorems are illustrated by numerical examples.
4
Content available Structurally R-controllable and structurally R-observable descriptor linear electrical circuits
Kaczorek Tadeusz, Borawski Kamil
Journal of Automation Mobile Robotics and Intelligent Systems
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2021
|
Vol. 15, No. 3
81--86
EN
Structurally R-controllable and structurally R-observable descriptor linear electrical circuits are investigated. Sufficient conditions are given under which the R- controllability and R-observability of descriptor linear electrical circuits are independent of their parameters.
5
Content available Positive electrical circuits with the chain structure and cyclic Metzler state matrices
Kaczorek Tadeusz
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2021
|
Vol. 69, nr 4
art. no. e137731
EN
The cyclicity of the state matrices of positive linear electrical circuits with the chain structure is considered. Two classes of positive linear electrical circuits with the chain structure and cyclic Metzler state matrices are analyzed. Some new properties of these classes of positive electrical circuits are established. The results are extended to fractional linear electrical circuits.
6
Content available remote Generalized Frobenius matrices and angles between them in analysis of linear electrical circuits
Kaczorek Tadeusz
Przegląd Elektrotechniczny
|
2019
|
R. 95, nr 9
13--19
EN
Generalized Frobenius matrices and their inverses are applied in analysis of the linear electrical circuits. The basic properties of generalized Frobenius matrices are analyzed. It is shown that if the state matrix of electrical circuit has generalized Frobenius form then its inverse system matrix has also generalized Frobenius form. The notion of an angle between state matrices of linear electrical circuits is proposed and its basic properties are investigated.
PL
Zaproponowane w tej pracy uogólnione macierze Frobeniusa oraz ich odwrotności zostały zastosowane w analizie liniowych obwodów elektrycznych. Zostały zbadane podstawowe własności tych macierzy. Wykazano między innymi, że macierze odwrotne uogólnionych macierzy Frobeniusa mają również postać uogólnionych macierzy Frobeniusa .Wprowadzono pojęcie kąta między macierzami stanu liniowych obwodów elektrycznych oraz zbadano ich podstawowe własności.
7
Content available Independence of the controllability and observability of resistances in linear electrical circuits
Kaczorek Tadeusz
Journal of Automation, Electronics and Electrical Engineering
|
2019
|
Vol. 1, No. 1
9--15
EN
Sufficient conditions are given under which the controllability and observability of linear electrical circuits is independent of their resistances. In some particular cases the observability depends only on the capacitances or inductances of the electrical circuits.
8
Content available remote Normal positive linear systems and electrical circuits
Kaczorek T.
Przegląd Elektrotechniczny
|
2018
|
R. 94, nr 6
1--7
EN
The notion of normal positive electrical circuits is introduced and some their specific properties are investigated. New state matrices of positive linear systems and electrical circuits are proposed and their properties are analyzed. It is shown that positive electrical circuits with diagonal state matrices are normal for all values of resistances, inductances and capacitances.
PL
W artykule zaproponowano pojęcie dodatniego obwodu elektrycznego oraz przeanalizowano specjalne własności dodatnich układów i obwodów elektrycznych. Wykazano, że dodatnie obwody elektryczne z diagonalnymi macierzami stanu są zawsze normalne dla wszystkich wartości rezystancji, indukcyjności i pojemności.
9
Content available Normal positive linear systems and electrical circuits
Kaczorek T.
Prace Naukowe Politechniki Śląskiej. Elektryka
|
2018
|
z. 1
7--20
EN
The notion of normal positive electrical circuits is introduced and some their specific properties are investigated. New state matrices of positive linear systems and electrical circuits are proposed and their properties are analyzed. It is shown that positive electrical circuits with diagonal state matrices are normal for all values of resistances, inductances and capacitances.
PL
W artykule zaproponowano pojęcie dodatniego obwodu elektrycznego oraz przeanalizowano specjalne własności dodatnich układów i obwodów elektrycznych. Wykazano, że dodatnie obwody elektryczne z diagonalnymi macierzami stanu są zawsze normalne dla wszystkich wartości rezystancji, indukcyjności i pojemności.
10
Content available Positive continuous – time linear electrical circuit
Piotrowska E.
Poznan University of Technology Academic Journals. Electrical Engineering
|
2018
|
No. 93
299--309
EN
The positive linear continuous-time systems by the use of the Caputo and conformable fractional calculus are analyzed. The conditions for internally positive systems are presented. For the established electrical circuit, a solution fractional order state-space equation was developed for system with Caputo and conformable fractional derivative (CFD) definition.
11
Content available Analysis of the positivity and stability of linear electrical circuits with state-feedbacks
Kaczorek T.
Poznan University of Technology Academic Journals. Electrical Engineering
|
2018
|
No. 93
21--36
EN
The analysis of the positivity and stability of linear electrical circuits by the use of state-feedbacks is addressed. Generalized Frobenius matrices are proposed and their properties are investigated. It is shown that if the state matrix of electrical circuit has generalized Frobenius form then the closed-loop system matrix is not positive and asymptotically stable. Different cases of modification of the positivity and stability of linear electrical circuits by state-feedbacks are discussed and necessary conditions for the existence of solutions to the problem are established.
12
Content available Positive and stable electrical circuits with state-feedbacks
Kaczorek T.
Archives of Electrical Engineering
|
2018
|
Vol. 67, nr 3
563-–578
EN
The analysis of the positivity and stability of linear electrical circuits by the use of state-feedbacks is addressed. Generalized Frobenius matrices are proposed and their properties are investigated. It is shown that if the state matrix of an electrical circuit has generalized Frobenius form then the closed-loop system matrix is not positive and asymptotically stable. Different cases of modification of the positivity and stability of linear electrical circuits by state-feedbacks are discussed and necessary conditions for the existence of solutions to the problem are established.
13
Content available Characteristic equations of the standard and descriptor linear electrical circuits of integer and fractional orders
Kaczorek T.
Acta Mechanica et Automatica
|
2018
|
Vol. 12, no. 3
209--216
EN
The problem of calculation of the characteristic equations of the standard and descriptor linear electrical circuits of integer and fractional orders is addressed. It is shown that the characteristic equations of standard and descriptor linear electrical circuits are independent of the method used in their analysis: the state space method, the mesh method and the node method. The considerations are illustrated by examples of standard and fractional linear electrical circuits.
14
Content available remote Positive linear systems and electrical circuits with inverse state matrices
Kaczorek T.
Przegląd Elektrotechniczny
|
2017
|
R. 93, nr 11
119--124
EN
The positive linear systems and electrical circuits with inverse state matrices are addressed. It is shown that: 1) If the eigenvalues of asymptotically stable matrix are located outside the unit circle then the eigenvalues of asymptotically stable matrix positive and located inside the unit circle; 2) If the eigenvalues of asymptotically stable matrix are positive and located inside the unit circle then the eigenvalues of the matrix Mn are located in the left half plane outside the unit circle. A new problem concerning the discretization of positive linear systems and electrical circuits is formulated and solved.
PL
W pracy przedstawiono dodatnie układy liniowe i obwody elektryczne z odwrotnością macierzy stanu. Zostało pokazane że: 1) Jeżeli wartości własne asymptotycznie stabilnej macierzy AMn są ulokowane poza obszarem koła jednostkowego wtedy wartości własne asymptotycznie stabilnej macierzy są dodatnie i leżą wewnątrz koła jednostkowego; 2) Jeżeli wartości własne asymptotycznie stabilnej macierzy są dodatnie i leżą wewnątrz koła jednostkowego wtedy wartości własne macierzy leżą w lewej półpłaszczyźnie poza obszarem koła jednostkowego. Sformułowany zostanie nowy problem dotyczący dyskretyzacji dodatnich układów liniowych oraz obwodów elektrycznych.
15
Content available remote Modification of the stability and positivity of standard and descriptor linear electrical circuits by state feedbacks
Borawski K.
Przegląd Elektrotechniczny
|
2017
|
R. 93, nr 11
176--180
EN
The modification of the stability and positivity of standard and descriptor linear electrical circuits by state feedbacks is investigated. It is shown that: 1) There is a class of nonpositive and unstable R, L, e circuits that can be stabilized and modified to positive ones by state feedback; 2) There is a class of nonpositive and stable R, L, e circuits that can be modified by state feedback to positive ones without loss of stability. The modification of stability and positivity of linear descriptor electrical circuits is addressed. Considerations are illustrated by examples of linear electrical circuits.
PL
W pracy rozpatrzono problem modyfikacji stabilności i dodatniości standardowych i deskryptorowych liniowych obwodów elektrycznych poprzez sprzężenie zwrotne od wektora stanu. Pokazano, że: 1) Istnieje klasa niedodatnich i niestabilnych obwodów typu R, L, e, które mogą zostać ustabilizowane i zmodyfikowane do obwodów dodatnich; 2) Istnieje klasa niedodatnich i stabilnych obwodów typu R, L, e, które mogą zostać zmodyfikowane do obwodów dodatnich bez utraty stabilności. Rozpatrywany problem uogólniono dla klasy układów deskryptorowych. Rozważania zilustrowano przykładami obwodów elektrycznych.
16
Content available remote Determinants of the matrices of solutions to the standard and positive linear electrical circuitsWyznaczniki macierzy rozwiązań standardowych i dodatnich liniowych obwodów elektrycznych
Kaczorek T.
Przegląd Elektrotechniczny
|
2017
|
R. 93, nr 2
278--283
EN
Determinants of the matrices of solutions to the standard and positive time-invariant and time-varying linear electrical circuits are addressed. Necessary and sufficient conditions for the positivity and asymptotic stability of the linear time-varying electrical circuits are established. It is shown that the determinants of the matrices of solutions to the standard and positive linear electrical circuits are nonzero and they decrease to zero for time tending to infinity if the electrical circuit contains at least one resistance and tends to 1 if the electrical circuit does not contain resistances. Determinants of the matrices of solutions of asymptotically stable electrical circuits tend to zero for time tending to infinity.
PL
Praca jest poświęcona analizie wyznaczników macierzy rozwiązań standardowych i dodatnich obwodów elektrycznych o stałych i zmiennych w czasie parametrach. Podano warunki konieczne i wystarczające dodatniości i stabilności asymptotycznej obwodów liniowych o zmiennych w czasie parametrach. Wykazano, że wyznaczniki macierzy rozwiązań standardowych i dodatnich obwodów o stałych i zmiennych w czasie parametrach są niezerowe i dążą do zera dla czasu dążącego do nieskończoności, gdy obwód elektryczny zawiera przynajmniej jedną rezystancję i dąży do 1, gdy obwód nie zawiera rezystancji. Wyznaczniki macierzy rozwiązań stabilnych asymptotycznie obwodów elektrycznych dążą do zera dla czasu dążącego do nieksończoności.
17
Content available remote CFE method – quality analysis of the approximation of reverse Laplace transform of fractional order
Zawadzki A., Włodarczyk M.
Prace Naukowe Politechniki Śląskiej. Elektryka
|
2017
|
z. 3-4
21--29
EN
The paper presents the CFE (Continued fraction expansion) method, which allows determination of inverse transform of expression containing element sα by expanding it into a continued fraction. On the basis of the presented method, analysis of selected electrical circuits containing quasi-elements described with fractional derivatives was performed. Calculations and numerical simulations were also carried out.
PL
W pracy przedstawiono metodę CFE (continued fraction expansion) umożliwiającą wyznaczanie transformaty odwrotnej wyrażenia zawierającego czynnik sα dzięki rozwinięciu w ułamek łańcuchowy. W oparciu o prezentowaną metodę przeprowadzono analizę wybranych obwodów elektrycznych zawierających quasielementy opisane pochodnymi niecałkowitego rzędu. Przeprowadzono obliczenia i wykonano symulacje numeryczne.
18
Content available Characteristic equations of the standard and descriptor linear electrical circuits
Kaczorek T.
Poznan University of Technology Academic Journals. Electrical Engineering
|
2017
|
No. 89
11--23
EN
The problem of calculation of the characteristic equations of the standard and descriptor linear electrical circuits is addressed. It is shown that the characteristic equation of the linear electrical circuit is independent of the method used for its analysis. The well-known three methods: the state space method, the mesh method and the node method are analyzed.
19
Content available Superstabilization of positive linear electrical circuit by state-feedbacks
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2017
|
Vol. 65, nr 5
703--708
EN
The concept of superstability of positive linear electrical circuits is introduced and its properties are characterized. The superstabilization of positive and nonpositive electrical circuits by state-feedbacks is analyzed.
20
Content available Specific duality and stability of positive electrical circuits
Kaczorek T.
Archives of Electrical Engineering
|
2017
|
Vol. 66, nr 4
663--679
EN
The specific duality and asymptotic stability of the positive linear electrical circuits are addressed. The specific duality of positive linear electrical circuits composed of resistances, inductances, capacitances and source voltages is established. 1) The linear electrical circuits are positive if and only if the common branches between meshes with resistances and inductances and meshes with resistances and capacitances contain only source voltages; 2) In linear electrical circuits the interchanges of the inductances by the capacitances and the capacitances by inductances do not change the asymptotic stability of the electrical circuits. The asymptotic stability of the positive and nonpositive electrical circuits is analyzed.
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